General correspondence for continued fractions (Q1107046)

The author extends the definition of general convergence for continued fractions with complex elements \(K(a_ n/b_ n)\)- which uses the notion of chordal metric d on the Riemann sphere - to the case where \(a_ n\), \(b_ n\) belong to an arbitrary field \({\mathbb{F}}\) and d is replaced by any bounded metric, say d, on \({\hat {\mathbb{F}}}={\mathbb{F}}\cup \{\infty \}\) which has the property that it leaves the cross-ratio of every set \(\{u_ 1,u_ 2,u_ 3,u_ 4\}\) of distinct elements from \({\hat {\mathbb{F}}}\) invariant under every non-singular linear fractional transformation T on \({\mathbb{F}}:\) \[ \frac{d(u_ 1,u_ 3)}{d(u_ 1,u_ 4)}\frac{d(u_ 2,u_ 3)}{d(u_ 2,u_ 4)}=\frac{\quad d(Tu_ 1,Tu_ 3)}{d(Tu_ 1,Tu_ 4)}\frac{d(Tu_ 2,Tu_ 3)}{d(Tu_ 2,Tu_ 4)}. \] It turns out that it is precisely this cross-ratio that is the fundamental tool in proving the important properties (called P1-P5 in the paper) of the notion of general convergence. Then the attention is focused on the definition of general correspondence. As ordinary correspondence of a continued fraction \(K(a_ n/b_ n)\) with elements in the field of meromorphic functions and a Laurent series can be seen as convergence in the well-known non-archimedean metric, measuring the order of contact in the origin, the author introduces general correspondence in a beautiful (natural) way, using an analogy of the chordal metric for pairs of (formal) Laurent series and the non-archimeden metric of the order of contact. This general correspondence is then shown to have the same nice properties P1-P5 as in the case of ordinary continued fractions, by proving the invariance of the cross-ratioö An interesting paper, both for theoretical and practical purposes.